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Schwartz kernel
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Definition
Let X 1 , X 2 X_1, X_2 be two open subsets of Cartesian spaces . Then a smooth function K ∈ C ∞ ( X 1 × X 2 ) K \in C^\infty(X_1 \times X_2) on the Cartesian product of these two manifolds defines a linear function
C cp ∞ ( X 2 ) ⟶ C ∞ ( X 1 ) ϕ ↦ 𝒦 ( ϕ ) ,
\array{
C^\infty_{cp}(X_2) &\overset{}{\longrightarrow}& C^\infty(X_1)
\\
\phi &\mapsto& \mathcal{K}(\phi)
}
\,,
by the “integral transform ”:
𝒦 ( ϕ ) : x 1 ↦ ∫ X 2 K ( x 1 , x 2 ) ϕ ( x 2 ) dvol ( x 2 )
\mathcal{K}(\phi) \;\colon\; x_1 \mapsto \int_{X_2} K(x_1, x_2) \phi(x_2) \, dvol(x_2)
More generally, this expression makes sense for
K ∈ 𝒟 ′ ( X 1 × X 2 )
K
\;\in\;
\mathcal{D}'(X_1 \times X_2)
a distribution on X 1 × X 2 X_1 \times X_2 (a “distribution of two variables”), which makes the result itself in general be a distribution
C cp ∞ ( X 2 ) ⟶ 𝒟 ′ ( X 1 ) ϕ ↦ 𝒦 ( ϕ ) .
\array{
C^\infty_{cp}(X_2) &\overset{}{\longrightarrow}& \mathcal{D}'(X_1)
\\
\phi &\mapsto& \mathcal{K}(\phi)
}
\,.
Here K K is called the integral kernel and 𝒦 ( ϕ ) \mathcal{K}(\phi) the corresponding integral transform .
Properties
Schwartz kernel theorem
The Schwarz kernel theorem states that this construction constitutes a linear isomorphism between Schwartz integral kernels and “distribution-valued distributions”
𝒟 ′ ( X 1 × X 2 ) ⟶ ≃ 𝒟 ′ ( X 2 , 𝒟 ′ ( X 1 ) ) K ↦ 𝒦
\array{
\mathcal{D}'(X_1 \times X_2)
&\overset{\simeq}{\longrightarrow}&
\mathcal{D}'( X_2, \mathcal{D}'(X_1) )
\\
K &\mapsto& \mathcal{K}
}
\,
Proposition
(partial product of distributions of several variables )
Let
K 1 ∈ 𝒟 ′ ( X × Y ) AAA K 2 ∈ 𝒟 ′ ( Y × Z )
K_1 \in \mathcal{D}'(X \times Y)
\phantom{AAA}
K_2 \in \mathcal{D}'(Y \times Z)
be two distributions of two variables . For their product of distributions to be defined over Y Y , Hörmander's criterion on the pair of wave front sets WF ( K 1 ) , WF ( K 2 ) WF(K_1), WF(K_2) needs to hold for the wave front wave vectors along X X and Y Y taken to be zero.
If this is satisfied, then composition of integral kernels (if it exists)
( K 1 ∘ K 2 ) ( − , − ) ≔ ∫ Y K 1 ( − , y ) K 2 ( y , − ) dvol Y ( y ) ∈ 𝒟 ′ ( X × Z )
(K_1 \circ K_2)(-,-)
\;\coloneqq\;
\underset{Y}{\int}
K_1(-,y) K_2(y,-)
dvol_Y(y)
\;\in\;
\mathcal{D}'(X \times Z)
has wave front set constrained by
WF ( K 1 ∘ K 2 ) ⊂ WF ( K 1 ) ∘ WF ( K 2 ) ∪ ( X × { 0 } ) × WF ( K 2 ) ∪ WF ( K 1 ) × ( Z × { 0 } ) ,
WF(K_1 \circ K_2)
\;\subset\;
WF(K_1) \circ WF(K_2)
\;\cup\;
(X \times \{0\}) \times WF(K_2)
\;\cup\;
WF(K_1) \times (Z \times \{0\})
\,,
where on the left the composition symbol means composition of relations of wave vectors over points in Y Y .
(Hörmander 90, theorem 8.2.14 )
References
Lars Hörmander , section 5.2 of The analysis of linear partial differential operators , vol. I, Springer 1983, 1990 (pdf )
See also
Last revised on November 9, 2018 at 11:14:25.
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